Stiffness of the basilar membrane.

["reinifrosch@xxxxxxxxxx" <reinifrosch@xxxxxxxxxx>]

Dear colleagues,

What is the stiffness S of the basilar membrane (BM) at the apex of the human
cochlea? The SI unit of spring constants is N / m (Newton per meter); the
stiffness S is a spring constant per BM surface unit, so the unit for S is N / m^3.

In the present case, fairly accurate results (+/- 30 percent or so) are obtained 
if the two-dimensional model shown, e.g., in Fig. 5.5 of Geisler (1998), "From 
Sound to Synapse", Oxford Univ. Press, is used: BM element dx = rigid beam 
on springs.

First result for S at x = 32 mm from base:

Fig. 11-73 of Bekesy (1960), "Experiments in Hearing". Pressure difference =
1 cm of water = 100 N / m^2 = 100 Pascal.
Scale on the right yields S = 1.4 * 10^7 N / m^3.
For an apical BM width dy = 0.5 mm, and for the element dx = 1mm mentioned
in the figure caption, the scale on the left yields S =  1.25 * 10^7 N / m^3, about
equal to the above-mentioned right-scale result.

As posted yesterday, I suspect that these stiffness results may be too large
because of the agar used to close the helicotrema (which is near the BM
at the apex).

Therefore, second result for S at x = 32 mm:

In Bekesy's Fig. 11-43, "passive" BM vibration patterns are shown. At 25 cps, 
the passive peak is not reached. At 50 cps, that peak is at x_pp = 32 mm 
and has a -3dB-width delta-x of about 9 mm. (The ordinate scales are linear). 
Calculations similar to those described in Section 5 of de Boer (1996), "Mechanics 
of the Cochlea", in "The Cochlea", Dallos et al., Eds., Springer, New York, have 
yielded that, at given frequency, the BM resonance place is apical of the passive 
peak by about the just mentioned width delta-x. At 400 cps, Fig. 11-43 yields
x_pp = 24 mm, delta-x = 8 mm. So the BM-resonance frequency at
x = 24 + 8 = 32 mm is concluded to be f_BMR = 400 cps. 

Estimated effective BM surface mass density M = 0.1 kg / m^2.
Formula for resonance frequency: f_BMR = [1 / (2pi)] * sqrt(S / M).
Resulting BM stiffness: S = M * (2pi * f_BMR)^2 = 6 * 10^5 N / m^3.

De Boer's exponential formula (caption of his Fig. 5.2) yields, at x = 32 mm,
S = 7 * 10^5 N / m^3, in good agreement with the just mentioned result.

These S-values are smaller than those based on Bekesy's Fig. 11-73
by a factor of ~(1 / 20).

Reinhart.

Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx .